a table of integrals of the error functions

JOURNAL OF RESEARCH of the Natiana l Bureau of Standards - B. Mathematical Sciences Vol. 73B, No. 1, January-March 1969

A Table of Integrals of the Error Functions*

Edward W. Ng** and Murray Geller**

(October 23, 1968)

This is a compendium of indefinite and definite integrals of products of the Error function with e lementary or transcendental functions. A s'ubstantial portion of the results are ne w.

Key Words: Astrophysics; atomic physics; Error functions; indefinite integrals; special functions; s tatistical analysis .

1. Introduction

Integrals of the error function occur in a great variety of applications, us ualJy in problems involv ing multiple integration where the integrand con tains ex pone ntials of the squares of the argu­me nts. Examples of applications can be cited from atomic phys ics [16),1 astrophysics [13] , and s tati s tical analysis [15]. This paper is an attempt to give a n up-to-date ex hau stive tabulation of s uch integrals.

All formu las for indefi nite integrals in sec tions 4.1, 4.2, 4.5, a nd 4.6 below were derived from integration by parts and c hecked by differe ntiation of the resulting expressions_ Section 4.3 and the second hali' of 4.5 cover all formulas give n in [7] , with om iss ion of trivi al duplications and with a number of additions; section 4.4 covers esse ntially formulas give n in [4], Vol. I, pp. 233-235. All these formu las have been re-derived and checked, e ithe r fro m the integral representation or from the hype rgeo metric series of the error function. Sections 4.7,4_8 and 4_9 ori ginated in a more varied way. Some formulas were derived from multiple integrals involvin g ele mentary fun ction s, others from existing formulas for integral s of conflue nt hypergeometric functions, and still others, a s mall portion, were compiled directly from existing literature. In connection with the last three sections, the reader should refer to [3] and [4], Vol. II, pp. 402,409-411.

Throughout this paper, we have adhered to the notations used in the NBS Handbook [9] and we have also assumed the reader's familiarity with the properties of the error functions, for whic h he is referred to [5]. In addition , the reader should also attend to the following conventions:

(i)z=x+iy=r exp (iO) is a complex variable,

~(z)=x, J1(z) = y, Iz l=r, argz = O;

(i i) the parameters a, b, and c are real and positive except where otherwise stated; (iii) unless otherwise specified, the parameters nand k represent the integers 0 , 1, 2 ... ,

whereas the parameters p, q, and v may be nonintegral; (iv) the integration constants have been omitted for the indefinite integrals; (v) when x is used (in stead of z) as the integration variable, it means that the formula has been

es tabli shed only for real x, though it may still be valid for certain complex values;

(vi) the integration symbol f denotes a Cauchy principal value.

*An invited pape r. This pape r presents the results of one phase of research carried out at the Jet Propu lsion Laboratory , California Ins titut e of Technology,

under Contract #NAS7- IOO. sponsored by the National Aeronautics and Space Administration. "Present address: Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif. 91103. 1 Figures in bracket~ indicate the lit erature references at the end of thi s paper.

1

A (x)

C(Z)

Ci(z)

DvCz)

e,,(z)

Ei(x)

IFI (a; b; z) == M(a, b, z)

2. Glossary of Functions and Notation

Gaussian Probability Function

Fresnel Integral

Cosine Integral -f '" cos t dt z t

Parabolic Cylinder Function

Truncated Exponential

Exponential Integral

Exponential Integral

n k

2~ k=O k!

f"'e- t -dt

z t

f'" e-t - -dt

- x t

Confluent Hypergeometric Function

kFi(al ... ak; b l ... bi ; z) Generalized Hypergeometric Function

H,,(x) Hv(x) Iv(z) Jv(z) Kv(z)

L~ Mp , q(z) Yv(z)

P(x)

(p )11

PI/(x) P~(z)

S(z)

si(z)

U(a, b, z) == 'I'(a, b, z) Wp,q(z)

Hermite Polynomial Struve Function Modified Bessel Function Bessel Function Modified Bessel Function Generalized Laguerre Polynomial

Whittaker Function Neumann Function (Bessel Function of Second Kind)

Probability Function

Pochhammer's Symbol f(p+n)/f(p) Legendre Polynomial Associated Legendre Function of the First Kind

Fresnel Integral

Sine Integral foo. dt - smt-

z t

Confluent Hypergeometric Function Whittaker Function

2

(ak)n zn

(b i )" n!

y f(p)

y(p , z)

f(p, z)

~(z)

!/J(z)

Euler's Constant Gamma Function

0.5772156649 ...

Incomplete Gamma Function

Incomplete Gamma Function

Riemann's Zeta Function

Psi Function d dz [In f(z)]

3. Definition and Integral Representations

3.1. Definitions and Other Notations

1. erf (z) == _2_ Ji!: e- /2dt, y;. 0

2. erfc (z ) == ...;; L" e-12dt= 1- erf (z ),

3. erfi (z) == - i e rf (iz) = ...;; r el 2dt.

Some authors use the above notations without the fac tor ...;;, and so me use <J:l(z) for erf (z).

4. w(z) == e- z2 erfc ( - iz ).

For real x, Dawson 's integral is defined as

5. F (x) == V; e- x2 e rfi (x)= V; Y [w(x)].

The error fun ction is al so closely related to the Gauss ian probability fun ctions:

6. erf (x) =2P(xV2)-1=A(x Y2).

3.2. Integral Representations

2az II 1. erf (az)= - - e- a'Z'I' dt y;. 0

2. erfc (az)=-2- e - a 2z2 ( ; - U'+ 2aztJdt y;. Jo

3. erf (az+~)= ~ exp (c- ::) f e- ((h' +2wz+c) dz.

3

- --------

( Z) 2a (Z2) i 00 4. erfc - =- exp c-- e-(a212+2zl+c)dt a ~ a2 0

5. e2ab erf ( ax + ~) + e- 2ab erf (ax - ~) =~ J e-a2x2- b2/x2dx

6. er c az = - e a z f ( ) 2a _ 22j "' e- a2(2tdt, ~(a»O,~(z»O.

~ 0 (Z2+t2)1 /2

2z (00 e-a2(2dt 7. e rfc (az) = 7r e- a2z2 Jo (t2 + Z2), ~(a) > 0, I arg zl < 7r, Z # O.

4 (I e-x212dt 8. 1- [erf (x)]2 =; e- x2 Jo (t 2 + 1)' x> 0

9. erf (x + in + erf (x - in = ~ ey2/4 J e- x2 cos xydx

10. erf (x+ in - erf (x- iJ) = i~ ey2/4 J e- x2 sin xydx

11. erf (x)=~ (" ex2 cos (J cos (x2 sin 0+ 0/2)dO , x # 0 V; Jo

12. erf (x)=.!.. ( '" e- I sin (2xVt) dt. 7r Jo t

4. Integrals of Product of Error Functions With Other Functions

4.1. Combination of Error Function With Powers

1. J erf (az)dz=z erf (az) + a~ exp (-a2z2)

2. J erfc (az)dz=z erfc (az) - a~ exp (-a2z2)

3. ( 00 erfc (ax)dx= ,II' Jo aV7r

4. J erf (az)zdz=~ Z2 erf (az) - 4~2 erf (az) + 2a~ exp (- a2z2)

5. J erfc (az)zdz=~ Z2 erfc (az) + 4~2 erf (az) - 2a~ exp (-a2z2)

( 00 1 6. Jo erfc (ax)xdx= 4a2'

4

r (~+ 1) f ZIHI e- a2z2 I - I 2 zn-2k 7. erf (az)zndz=--=t=l erf (az) + ,I L ( ) ~

n aV7T(n+l) k =O r ~-k+l a 2

I-j r (l+4) --+ 1 ,I erf (az),

n a,,+1 V7T j = 0 or 1, 2l - j = n + 1

f (n+2)(n+l)f ( n+2) Zll+1

8. erf (az)z1l+ 2dz= 2(n+3)a2 erf (az)zndz+ Z2- 2a2 (n+3) erf (az)

r (~+ 1) 9. f erfc (az)zndz= (z:'l) erfc (az) _ ,~a2z2 ~ _(-,-2_-,-_) Z:~:k

n a v 7T (n + 1) k = O r ~ - k + 1 2

1-j r (l+~) +--1 . ,I erf (az),

n+ an+lv7T j=O or 1, 2l - j=n+l

f (n+2)(n+l)f ( n+2) zn+1

10. erfc (az)zn+2dz= 2(n+3)a2 erfc (az)z"dz+ Z2 - 2a2 (n+3) erfc (az)

" r (~+ 1 ) 11. { erfc (ax)xndx= v:;;. . ' Jo (n+ 1) 7Tan+1

12.2 f erf (az)z - 'dz= In z erf (az) - ~ f In Ze- 0 2Z 2dz

13. f erfc (az)z-'dz= In z erfc (az) + ~ f In ze- a2z2dz

f erf (az) 2a f 1

14. erf (az)z -ndz=- . + - e- a2z2dz (n-1)zn- 1 (n - l)v:;;. zn- I '

f erfc (az) 2a f 1 15. erfc (az)z-"dz=- - - e-a2z2dz (n-l)zn- , (n-l)v:;;. zn-I '

n~2

f ZP+I 1 (p

16. erf (az)zPdz=--l erf (az) - ,/"Y -2+ 1, a2z2) , p + (p + 1) aP+ I V 7T

p>-2,p~-1

2 See append ix for integrals on the right-hand sides of eqs (12 to 15).

5

------

17. J erfe (az)zpdz= zP+11· erfe (az) + 1 v:;;. 'Y (E2 + 1, a2z2 ) ,

p + (p + 1) aP+ 1 7r

18. [ 00 erfe (ax)xpdx= 1 v:;;. r (E2 + 1), Jo (p + l)ap+1 7r

7r largal < 4",p > -l

foo al - p (p) 19. erf (ax)x p- 2dx= , I r -2 '

o V7r(l-p)

7r larga l < 4"'o < p<1.

p >-1

4.2. Combination of Error Functions With Exponentials and Powers 3

1. J erf (az)ebzdz=i ebz erf (az)-i exp (:;2) erf (az- ;a)

2. J erfe (az)ebZdz=i ebz erfe (az)+i exp (:;2) erf (az- ;a)

3. 100 erf (ax)e- bXdx=i exp (:;2) erfe (2bJ, PJ(b) > 0, I arg al < 7r/4

5. J erf (az)ebZzdz=1: erf (az)ebz (z-1:) -1: exp (~) { (~_1:) erf (t)- _1_ e- t2 }, t= az-!!"'-b b b 4a2 2a2 b a v:;;. 2a

6. J erfe (az)ebZzdz=i erfe (az)ebz (z-i)

1 ( b2 ) { (b 1) 1 2} b +- exp - --- erf (t)- -- e- t ,t= az--b 4a2 2a2 b a v;: 2a

[ 00 1 ( b2 ) [1 b] ( b ) 1 7r 7. Jo erf (ax)e -bxxdx=b exp 4a2 b- 2a2 erfe 2a + abV;:' PJ(b) > 0, larg al <4"

J nl 9. 4 erf (az)ebzz"dz=(-I)n bn~l ebz erf(az)en(-bz)

2a n 1 n (- b)k J -- (-1)" _. ,,-- Zk exp (- a2z2+ bz)dz v;: bn+1 f:o k!

3 1n this section a and b ca n take any value on the co mplex plane olher than the origin , except where otherwise stat ed .

.. See appendix for the integrals on the right -hand sides of eqs (9 to 12).

6

10. b I erf (az)ebZz"dz+ n I erf (az)ebZz"- l dz

=ebzzn- I [z erf (az)+_1- e- 02z2 ] a v:;;.

11. I erfc (az)ebZz"dz = (-I )n b:;{ ebz erfc (az)e,,( - bz)

2a ( - I)"n ! II (-W I ?'J + - '" -- Zk exp ( - a-z- + bz)dz y:; b,,+1 to k !

12. b I erfc (az)ebZz"dz + n I erfc (az)ebzzn - 1dz

= ebzz,,- I [z erfc (az) __ 1_ e- U2Z2 ] a v:;;.

+ I (bz" + nz"- I) exp (- a2z2 + bz)dz

13. (oo e rf(ax) e-bXxlldx=(;) 1/2 ± ~~~ I (2a/~ exp (:~2) D- II +k - 1 ( ~ ~) Jo k =O a v 2

_ 11 (_ 1)II - k n!(a)k- n d" - k [ (q2) .. (9.) ] - k~ bk+ 1 (n - k)! dq"- k exp 4 ed c 2 '

14. (00 erfc (ax)ebxx"dx = (-1)'1+ 1 bn ! Jo n+1

b q=-,

a Y2 (b) > 0,

( 2) 1/2 II n! ~!! (b 2 ) (b ) + -; ~ (_I)k bk + 1 (2a2 ) 2 exp 8a2 D - II +J.- - I - a v'2

1T larg al < -

4

1T la rg (b-a)1 < ";r

4 .3. Combination of Error Function With Exponentials of More Complicated Arguments

1a v:;;. 1. e- x2 erf (x)dx=- (erf a)2

o 4

100 V 1T 1 b 2. erf (ax)e - b2x2dx=-- - - tan- I -

o 2b b v:;;. a

('" 1 [a + bJ 3. Jo erfc (ax) eb2x2dx = 2 y:;;. b In a - b ' b may be complex,

7

1T larg al <-

4

6 ('" f ( ) -b'x' 2d _ V; __ 1_ [~ _ I ~ _ a ] I I 1T . Jo er axe x x- 4b3 2V; b3 tan a b2(a2+b2) ' arga <"4

7 f "' f ( ) - b'x' 2d __ 1 _ [~ _ I ~ _ a ] I I 1T . Jo ere ax e x x- 2 V; b3tan a b2(a2+b2) ' arga <"4

1'" a (p) (1 p 3 a2) 8. erf (ax) e-b'x'xpdx = - b- p- 2f - + 1 oF I - - + l' _. --o V; 2 - 2' 2 ' 2' b2 '

9t(p) > -2

9. foo erfe (ax) eb'x'xpdx = f(tp + 1) .,F (P + 1 P + 2. p + 3. b2) Jo V; (p + 1)aP+1 - I 2' 2 ' 2 'a 2 '

10. r erfe (x)eX'xp-ldx=~ sec (P21T) r (~), O< p < 1

Ix 1 ( b2 ) (b2

) 12. erf (iax)e -u'xLbxdx=. exp - Ei -- , o 2m .,,;;. 4a2 4a2

.92 (b) > 0,

foo V; ( b ) 13. erf (x) e- (ax+b)'dx = - - erf , -oo a Ya2 + 1

foo i [1 a (a2) (a2

)] 14. Jo erf (ix)e-(x2+ux)xdx= V; -;;-+4 Ei -"4 exp "4 ' .92 (a) > 0

16. e rfc (ax)eU2Xx - 3dx=- (l-2a2 )eQ ' erfe (a) +-f'" 1 a I 2 V;

f'" (X - l)p/2- 1 1 (p) 17. J I erfc (ax) ea'x xp+1 dx =.,,;;. ea'/225P/4-2aP/2- 1 r 2" D_1_p (a v2) , p > O

O< p < 1

8

,?4>(a) > 0, 22. l oo erf (~) e-b2x2xdx = 2~2 (1- e-2ab ) ,

23 (00 erfc (~) e-b2x2xdx = _1_ e-2ab . Jo X 2b2 ' ,?4>(a) > 0,

(00 (a) dx 24. Jo erfc ~ e- b2x2 -;=-Ei(-2ab), ,?4> (a) > 0,

f'''' - b2 1 b2

25. Jo erfc (ax)e wxdx=4a2e- ab(l+ab)-4 [-Ei(-ab))

fOO - !!:.. dx 26. erf (ax)[l-e 4X2] -=y+ln (ab)+ [-Ei( - ab)]

o x

foo _ 1J2 dx 27. e rfc (ax)e X2 -=-Ei(- 2ab) ,

o x ,?4> (a) > 0,

a>O

p>o

30. t erfc (D exp (:2 - b2X2) dX=b~ [sin 2bCi(2b)-cos2bsi(2b))

I arg bl <~, ,?4>(a) > ° · x ( b) 34. L erfc ax +:;: e-c2x2xdx

= t(a2 + c2)- 1/2[a + (a2 + c2)1 /2] - 1 exp [ - 2b(a + Va2 + c2)],

f'" { (b - 2ax2) (b + 2ax2) } 1 35. 2 cosh ab - caberf - ealJ erf e-(cL a2)x2xdx = -- e- bc, o 2x 2x c2 - a2

a > 0, b > 0, .<n(c2) > ° 9

36. i "" cosh (2bx) exp [(a cosh X)2] erfc (a cosh x)dx

9P(a) > 0, -t < 9P(b) < t

37. 100 {exp [ - (x - a)2] - exp [ - (x + a)2]) erf (x )dx = -v;. erf (a/V2)

100 -v;. 38. 0 {exp [ - (x - a)2] + exp [ - ~ + a)2]) erf (x )dx=-2- {I + [erf (atV2))2}·

4.4 Definite Integrals From Laplace Transforms Involving Erf (Vax)

2. 1"" e-bx[eax erfc (~)dx]=b- 1 /2(v'b+Va)- 1

3. Lx> e-bx[eax erf(~) -2(aX/7T)1 /2]dx=(b-a)-I(a/b)3/2, b >a

lox , ~ (v'b- Va)2 9. e-bx[8axe ax erfc (vax)-8(ax/7T)1/2+1]dx=b- 1 o v'b+Va

10

17. e-bx 2 - e- a2/(4X)-aerfc -- dx=--' 1'" [ (X)1/2 ( a )] e-av "/j

o 7T 2Vx b3/2

4.5. Combination of Error Function With Trigonometric Functions

1. J erf (az) sin bzdz=-i cos bz erf (az) + 21b exp ( - :~2) {erf (az- i ;a) + erf (az+ ~:) }

2. J erf (az) cos bzdz = i sin bz erf (az) + 2ib exp ( - :~2) {erf (az - ~:)- erf( az+ ~:) }

3. J erfc (az) sin bzdz=-i cos bz e rfc (az) - 21b exp ( - :~2) {erf (az - ;:) + erf ( az + ;:) }

4. J erfc (az) cos bzdz=i s in bz erfc (az)- 2ib exp ( - :~2) {erf (az- ;:) - erf (az+ ;:) }

s. 10'" erfc (ax) sin bxdx=i [l-exp (- :~2) ], 7T larg al <-

4

('''' i (b2) (ib) 6. Jo erfc (ax) cos bxdx=-{; exp - 4a2 erf 2a '

7T larg al < 4"

foo l (b2 ) 1 [ (b2

) ] 7. 0 erfc (ax) cos (bx}xdx= 2a2 exp - 4a2 - b2 1- exp - 4a2

8. erfc (~) sin bxdx=-- -- [(a2 +b2)1/2-a]-1/2, Joe 1 (a/2 )1 /2 I) b a2 + b2

9t(a) > 1J1(b)1

9 erfc (~)\ cos bxdx = -- r (a 2 + b2 ) 1/2 + a] - 1/2 J'" ( a/2 )1/2 . 0 a2 + b2 L ,

9t{a) > 1J1(b)1

11

r (P+3) b 11. (oc erfc (ax) sin bxxpdx = 2 2F., (p+2 p+3. ~ P+4._~)

Jo aP+2Vrr(p+2) - 2' 2 ' 2' 2 ' 4a2 '

PJ2(a) >O,.92(p) >0

12. ( '" erfc (ax) cos bxxIJdx = r (~+ 1) .,F2 (p+ 1 p+2.1:. p+3. Jo aIJ+1Vrr(p+I) - 2' 2 ' 2' 2 '

PJ2 ( a) > 0, .92 (p) > 0

J oc cos bx 1 [ . ( b2 )] 14. erf (ax) -- dx=- -Et --o x 2 4a2

( " cos px 1 { ( p2 ) (p2 )} 15. Jo [erfc (ax)-erfc (bx)] -x- dX=2 Ei - 4a2 -Ei - 4b2

16. fo" erfc ( ~) sin bxdx=i exp [- (2ab) 1/2] cos [(2ab) 1/2], .92(a) > 0, .92(b) > 0

17. f" erf (~)cos bxdx=-1;exp [-(2ab)1 /2] sin [(2ab)1 /2], .92(a) >0, .92(b) >0

tOO "(-1)1' (k2) 18. erfc(ax) tan xdx = L -k- exp-2 + In 2 o k = 1 a

20 f'''. ( ) . b· dx 1 {[ E· ( (e-b)2)] [ E · ((e+b)2)]} e--'--b . J 0 eri ax sm x sm ex ~="4 - L - ~ - - t - ~, ...-

21. fo" erfc(ax) sin bx sin ex ~=~ In I~~~I-f" erf(ax) sin bx sin ex ~, c *b

( " dx 1 {[ ((e-b)2)] [ ((e+b)2)]} 22. Jo erf(ax) cos bx cos eX~="4 -Ei - 4a2 + -Ei -~ , e*b

( '" Vrr ( b2 ) ( b ) 23. J 0 erfc (ax) sin bxea2x2dx= 2a exp 4a2 erfc 2a

(" iVrr (b2 ) 24. J 0 erf (iax) sin bXe- U2X2dx = ~ exp - 4a2

25. t, erfc (ax) cos bxeu2x 2dx= 2a~ exp (:;2) [ -Ei ( - :;2)]

26. foX erfc (ax) sin x cosh xdx = ~ [sin (2~2) - cos (2~2) + 1 ]

12

27. t" erfc (ax) cos x sinh xdx = ~ [ sin (2~2) + cos (2~2) -1 ]

28. t" erfc (ax) x cos x cosh xdx= [2~2 cos (2~2) -~ sin (2~2) ]

29. t' erfc (ax) x sin x sinh xdx= D cos (2~2)+ ~2 sin(2~2) -~]

30. I X [e- bxerfc (ab- 2X )-ebxerfc (ab+ 2

X )] sinpxdx= ( 22p b2) exp [-a 2(b2+p2)]. o a a p +

4.6. Combination of Error Function With Logarithms and Powers

1. J erf (az) In zdz= (In z-l) [z erf (az) + ,I; e- a2z2] - ~ ;Ei(- a 2z 2 )

a v 7T 2a v 7T

2. J erfc (az) In zdz = (In z -1) [z erfc (az) - ,I; e- a2z2] + : ; Ei(- a 2z2) a v 7T 2a v 7T

3·t" erfc(ax)lnxdx=- a~[I+~+Ina]

4. 5 J er[(az)zlnzdz= .!. Z2 erf (az) In z+ : ;z In ze - a'z' 2 2a V7T

_.!. J z erf (az)dz-~ erf (az) - _ 1_ J In ze - a2z'dz 2 4a- 2a V;

5. J erfc (az)zInzdz =~ Z2 erf (az) In z- 2:I':j;e - a2z,

-~ f z erfc (az)dz+ 4~2 erf (az) + 2a ~ J In ze - a2z2dz

6. J '" erfc (ax)x Inxdx=-I_+_I-J '" Inxe - a' x2dx o 8a 2 2aV; 0

7. (k + 1) J erf (az)zk In zdz = Zk +l erf (az) In z +_I_zk In ze - a2z2 a v;.

-f Zk erf (az)dz __ I_J zk - Ie- a2z2dz aV;

__ k_Jzk- I In ze - a'z2dz aV;

8. (k + 1) J erfc (az)zk In zdz= Zk+l erfc (az) In z __ I_zk In ze - a2z2 aV;

_JZk erfc (az)dz+_I-Jzk-le-a2z2dz aV7r

~ For th e elementary integrals in eqs. (4 to 8), see appendix.

13

9. (k+ 1) fox erfc (ax)x k Inxdx= 2~~:+1 [k~ 1 +~ '" (~) -k In a l 4.7. Combination of Two Error Functions

J ~ 1 (~ 1. erf (bx) erfc (ax)dx= • / o bV1T a

2. ( x erfc (bx) erfc (ax)dx= : / (a+b-v'a2+b2) Jo abv1T

3. t" erf (~) erfc (ax)dx

J'" (a) dx 1 2b. = erf (bx) erfc - 2=-- (1- e- 2ab ) -- [- EL(- 2ab)] o x x aV; V;

4. L" erfc (~) erfc (ax) dx

J'" (a)dx 1 2b. = erfc (bx) erfc - 2=--e- 2ab+- [-EL(-2ab)] o x x aV; V;

5. ( x erfc (ax) erf (bx)eb2x2dx= ~ ~ In (1- b:), a2 > b2 J 0 2b V1T a

6. to erfc (ax) erfc (bx)eb2X2dx= b~ In (1 +~), a+ b > 0

7. f erf(x) erf(v'1-x2)xdx=~(~-1)

=_1_ {e - ab[- Ei(-bc+ ba)] - eab [ - Ei(- bc- ba)]}. aV1T

4.8. Combination of Error Function With Bessel Functions

1. LX erf (ax)]o(bx)dx=i erfc (2:)

2. t, erfc (ax)}o(bx)dx=i erf (2ba)

14

L

3. f ' [2 e rfc (x)-I] ]n(bx)dx =- i [2 erfc (~) - 1]

4. Loo e rf (x)],(bx)dx=i e-b2/8/() (~)

S. J,"" erf (~) l:!iz(bX)X - I /2dx=b - 3/2 erf (~)

1 -1 < p < -

2

p >- 1

{'" . 1 (b)P 1 r (p + ~ ) (3 b2 ) B. In erfc (ax)],J(bx)xp+'dx= 2V;"2 a2p+2 f(p +2) ,F, P +"2; p+2; - 4a2 '

r ex> ., P _ 2- (p+2v+\}b"f(p+ v+ 1) 10. J( eIfc (ax)]v(bx)x dx - ( + V+3)

() aP+V+ I f ( v + 1) f "-P----:_..:c. 2

1 p > -

2

.F. (P+V+l p+v+2. +1 P + V+3. _ .iC) x 2 1 2 ' 2 ' v , 2 ' 4a2'

11. f''' erfc (ax)]()(bx)e(J':r'xdx = ~ I [1 - v; (~) eb' /4a' e rfc (~)] J() ab v 7T' 2a 2a

p+v > - 1

1"" 1 (b) P 1 (3 ) ( b2 ) 12 erfc (ax)] (bx)ef/'X'xP+'dx = - - - - r p + - eb' /4a' f - p - l -

. () P 27T' 2 a2p+2 2 '4a2 '

__ 1 _1 r ( +.!) b2/8a'W (.iC) - V; baP p 2 e - p/2.p/2 Ba2 '

14. r oc erfc (ax)] (bx)ef/'.r'xP+ 'dx=~ (!!.)P_l_ r (p+~) eb'/4a2f (_p_.! .iC) J() p 27T' 2 a2p+2 2 2' 4a2 '

1 -1 < p <-

2

p > -1

1 -1 < p < -

2

3 -1 < p < -

2

1 -1<p<-

2

16. L" erfc (x) Yp (bx)eX 'xP+3dx =- b~ f(p+2)e b' /8W_(P +3J/2,P/2 (~) , 3 -2 < p < - -

2

IS

1 --<p<O

2

1 1 -- < p < -

4 2

( 3) f 2p +"2 f (- p) 1 3 3 2 = 2/J-2a- 4J! - 2 F (P+ - 2p+ -' p+-' 1- - )

( 1) 2 1 2' 2' 2' a2 ' 1T p+ -

2

1 P o;f- -

2' -1 < !n (p) < 0

1 1 1 - - < p < -

2 2

21. L OO erfc (x)}HV(ax)}A _V(ax)x /Jdx

. ( 1 ) (1 pI p 3 p ) a 2Af A+ 2 P+l 4F4 I + A'2 + A, I + A+2'2+ A+2; I+A+v, I +A-v, 1+2A' 2 +A+2; -a2

V;:22A+ 1f(A+ v+ l)f(A- 1'+ 1) (A +4 p + 4)

1 A+ "2 p >O,1+2Ao;f-n.

4.9. Combination of Error Function With Other Special Functions

roo [.( 1 )]dx 00 (-a)k+1 1. Jo erfc (ax) -El -4 X2 --.;- = (y+ ln a)2+~(2)+2 ~ok!(k + 1P

2. r oo erfc (ax) [- Ei (_b2)] d~ = _I_ (1-2ab)e - 2f1b+ 2a2 [-Ei(-2ab)] Jo x2 Xl 2b2

3. r oo erfc (x)si(2px)dx= (e - I/4f12-1) _ v;: erfc (~) Jo 2a 2a

4. LX. erf (ax)Ci(x) ;=-~ [~(2) + (y-ln 4a2 )2] --i f (- 4~2)h'+ 1 k!(k~ 1P k=O

_2/L - 1 /L- :l/2 fl2/4f (IL+V+1) f (IL-V) W 2 - 1T a e 2 2 (1- 2/L)/4, (1 +2/L )/4(a),

IL < 1, IL < v; ~+v > -l, ~-V o;f -2n

16

7. JX erfc(x)L~J)(ax2)e-(a-!)X2x2P+ ldx o

p>-I

II-al < 1, p > -l

1 9 j "'erf(px)IFI(a'b.- p2X 2)x2(b- l)dx= r(b) . 2F1(a b -!.' b+1:..- 1 ) . 0 ' , V;p2b- I(2b-J) · , 2' 2' ,

b > .! a < -2' 2

10. erf(px) IF1(a' b' _ q2X2)X2(b - l)dx = .. F (a b- - ' b+-'-) f'" . r( b ) 1 1 - q2 o ' , V;p2/i-1(2b-l); · I , 2' 2' p2 '

b > .! 2' p2 > q2

r(!. v+ 1) _ 1__ 2 .F. ( v+2 v+ 1. b v+3 . _ q2)

:1 2 a, , " , , p,,+IV; (v+ 1) 2 2 2 p2

p ~ O, v >- I,

1 f(b ) (1 1) = 2p b (a-b)f(a) £1 2' b; a; I-j; ,

(p+l)(1-1 /2 [ 3 q ] = .,F 1 1, a; - ; , 2p (p + 1 - q) a - 2 p (p + 1 - q)

p ~ 0, p+ 1 ~ q, 92( p) > 92(q)

x f(2b)f(a-b+4)

14·10 erfc (x)IF1(a;b;-px2)ex2x2b- Idx = V;2 2b f(a+I) 2F1(a;b+i;a+ I; I-P), 1

92(b) >0, 92(b - a) <2' Il-pl < l

15. L' eel (x),F,(a; b; l-x'),"(I-x')'-'xdx~ ((b) 3) ,F, (a+l; b+~; I) , o 2f b+2

92(b»O

17

327-0 \ 6 0-69-2

=!! q2P+ le a2q2 f(p) F [1 ' +:J . 2(b 2)J 2 ( 3) I I , P "2, q - a , f P+2

p??-1

17. J: e/fe (~) D.(±x)e""dF 2~:':' [r (Y) + r t~)l .9t( v) oF- -1 ; .92( v) > -1 for lower sign

.lZ + ')

22 -a - :J/2 (1 1 1) 2F, 2 v + 1,2;-2 v + 2; l- -a ,

(V7T )

_ f( 4v + 1) f(,u -v +t) aV+I/2

- f(,u+ v+ 1)24V+ 1

la-ll < l,

20. LC erfc (x)M/L. v(ax2)e - I/2(a - 2)x2x2vdx

.9t( v) > 0

1 .9t(v-,u) <-.

2

f(2v+ 1)f (2V+~) f(,u-V)pV +I/2 ( 1 3 3)

( 3) 2F, ,u+v+ 2 '2+ 2v ; ,u+v+ 2 ; I-a , 27Tf ,u+ v+ 2

la-ll < l , .9t (,u) > .9t( v»-~

21. L'" erfc (x) ,M A.!L(ax2 )x JJ exp (ax2/2) ,dx

_ f(,u+~ p+~) a!L + 1/2

- 2 V; (,u + ~ p + 1)

5. Appendix. Some Relevant Integrals Involving Elementary Functions

f /I n! Zlt - k (AI) zlteazdz =eaz l: (_1)h' . -

k = O (n - k)! aN I

(A2) f e-a2z2+bzdz= V; exp (~) erf (az-~) 2a 4a2 2a

18

. I

(A3) ( '" ra2x2+bxdx = y:;;. exp (~) [1 + erf (!!...)] Jo 2a 4a2 2a

J 2 1 ( b2 ) [b y:;;. ( b ) ] (A4) ze- a z2+vzdz=- exp - -- erf az-- _e-(az-b/2a)2 2a2 4a2 2a 2a

1'" 1 [y:;;. b ( b) ] (AS) xe- a2x2+bxdx=- -- eb2/4a2 erfc -- + 1 o 2a2 2a 2a

J (b2) n n' ( b )"-k J (A 7) z"e-a2z2+bzdz = a-1l - 1 exp - '" . - uk e- u2du , 4a2 6 k!(n-k)! 2a

b u=az--

2a

f (k+ 1) e- u2 r - I 2 1 . 1

(A8) J uh-e- u2du=-- L U h-- 2j - I+- O- s )f(r+ - ) erf (u), 2 j = O f (k; 1 _ j) 2 2

k = 2r-s, s = 0 or 1

n e ven positive integer

n > 1 odd positive integer

where u = a (z + ;)

(A12) (z- a)"e-C2(Z-b)2dz = '" n. u"e- u2du J n' (b-a)n - k J k~ k! ( n - k) ! c" + I '

where u=c(z- b)-

1 (A14) aB1(p, a, u)=pBI(p-1, a, u)+ aP y(p, au)

( '" dx 1 00 ( - au) h-+ I

(A1S) Ju In xe- ax ~=2 {(y+ In au)2+ ~(2) + 2 In uE1 (au)) + 6 k!(k+ 1)3

(A16) ( '" xPe-ax In xdx= f(p+ 1) [ljJ(p+ 1) -In a] Jo ap+1

19

6. References

[1] Bhattacharya, S. K. , Proc. Camb. Phil. Soc. 63, 179 (1967). [2] Bierens de Haan , D., Nouvelles Tables d'Inll~grales Defines. (New York, G. E. Stechert and Company, 1939.)

[3] Bock, P., Composito. Mathematica J . 7, 123, (1939). [4] Erdelyi , A., Magnus, W. , Oberhettinger, F ., and Tricomi, F. G., Higher Transcendental Functions, Vol. I and 2 (Mc·

Graw·Hill Book Co., Inc., New York 1953). Tables of Integral Transforms. Vols. I and 2 (McGraw·Hill Book Co., Inc., New York 1954).

[5] Gautschi, W., Chapter 7 of [9] below. [6] Geller, M., A Table of Integrals Involving Powers, Exponentials, Logarithms and the Exponential Integral. Jet Pro·

pulsion Laboratory Technical Report No. 32-469(1963). [7] Gradshteyn, I. S., and Ryzhik, I. W., Tahle of Integrals, Series and Products. (Academic Press, New York 1965),

pages 648-653. [8] Grobner, W., an.d Hofreiter, N., Integraltafeln (Springer·Verlag, New York 1965). [9] Abramowitz, M. and Stegun, I. A. (Editors), Handbook of Mathematical Functions, NBS AMS 55, (Government Print·

ing Office', Washington , D. C. 1965). [10] Jahnke, E., Emde, F., and Losch, F. , Tables of Higher Functions (McGraw·Hill Book Co., Inc. , New York 1960). [11] Magnus, W., Oherhettinger, F., and Soni, R. P. , Special Functions of Mathematical Physics (Springer·Verlag, New

York 1966). [12] Mangulis, V., Handbook of Series (Academic Press, New York 1965). [13] Ng, E . W., Astrophysical 1. ISO, 787 (1967). [14] Short, P. 1. , Rev. Soc. Indust. Appl. Math. 7,565 (1965). [15] Zelen, M. and Severo, N. c., Chapter 26 of [9] above. [16] Zimering, S., 1. Mathematical Phys. 6, 336 (1965).

20

(Paper 73Bl-281)